ISI 2019 | PCB Mathematics | Question: 5

Question

Suppose $f$ and $g$ are continuous real valued functions on $[a, b]$ and are differentiable on $(a, b)$. Assume that $g^{\prime}(x) \neq 0$ for any $x \in(a, b)$. Prove that there exists $\xi \in(a, b)$ such that $$ \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)} $$

Answer 1

Lagrange's Mean Value Theorem - Given a function f that is continuous over [a,b] and differentiable over (a,b), then there exists a value c that belongs to [a,b] such that f'(c) = (f(b) - f(a)) / (b - a).

 

The first sentence of the question gives the pre-requisites for the above theorem. Therefore, the above theorem can be applied here.

 

f'(E) = (f(b) - f(a)) / (b - a) --> Eqn. 1

g'(E) = (g(b) - g(a)) / (b - a) --> Eqn. 2

 

As g'(x) is != 0 when x belongs to (a,b), we can divide Eqn. 1 with Eqn. 2.

 

Therefore we can write - f'(E) / g'(E) = (f(b) - f(a)) / (g(b) - g(a))